/*
 * @(#)Math.java	1.57 03/01/23
 *
 * Copyright 2003 Sun Microsystems, Inc. All rights reserved.
 * SUN PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
 */

package java.lang;
//KML import java.util.Random;


/**
 * The class <code>Math</code> contains methods for performing basic
 * numeric operations such as the elementary exponential, logarithm,
 * square root, and trigonometric functions.
 * <p>
 * Unlike some of the numeric methods of class
 * <code>StrictMath</code>, all implementations of the equivalent
 * functions of class <code>Math</code> are not defined to return the
 * bit-for-bit same results.  This relaxation permits
 * better-performing implementations where strict reproducibility is
 * not required.
 * <p>
 * By default many of the <code>Math</code> methods simply call
 * the equivalent method in <code>StrictMath</code> for their
 * implementation.  Code generators are encouraged to use
 * platform-specific native libraries or microprocessor instructions,
 * where available, to provide higher-performance implementations of
 * <code>Math</code> methods.  Such higher-performance
 * implementations still must conform to the specification for
 * <code>Math</code>.
 * <p>
 * The quality of implementation specifications concern two
 * properties, accuracy of the returned result and monotonicity of the
 * method.  Accuracy of the floating-point <code>Math</code> methods
 * is measured in terms of <i>ulps</i>, units in the last place.  For
 * a given floating-point format, an ulp of a specific real number
 * value is the difference between the two floating-point values
 * closest to that numerical value.  When discussing the accuracy of a
 * method as a whole rather than at a specific argument, the number of
 * ulps cited is for the worst-case error at any argument.  If a
 * method always has an error less than 0.5 ulps, the method always
 * returns the floating-point number nearest the exact result; such a
 * method is <i>correctly rounded</i>.  A correctly rounded method is
 * generally the best a floating-point approximation can be; however,
 * it is impractical for many floating-point methods to be correctly
 * rounded.  Instead, for the <code>Math</code> class, a larger error
 * bound of 1 or 2 ulps is allowed for certain methods.  Informally,
 * with a 1 ulp error bound, when the exact result is a representable
 * number the exact result should be returned; otherwise, either of
 * the two floating-point numbers closest to the exact result may be
 * returned.  Besides accuracy at individual arguments, maintaining
 * proper relations between the method at different arguments is also
 * important.  Therefore, methods with more than 0.5 ulp errors are
 * required to be <i>semi-monotonic</i>: whenever the mathematical
 * function is non-decreasing, so is the floating-point approximation,
 * likewise, whenever the mathematical function is non-increasing, so
 * is the floating-point approximation.  Not all approximations that
 * have 1 ulp accuracy will automatically meet the monotonicity
 * requirements.
 * 
 * @author  unascribed
 * @version 1.57, 01/23/03
 * @since   JDK1.0
 */

public final strictfp class Math {

    /**
     * Don't let anyone instantiate this class.
     */
    private Math() {}

    /**
     * The <code>double</code> value that is closer than any other to
     * <i>e</i>, the base of the natural logarithms.
     */
    public static final double E = 2.7182818284590452354;

    /**
     * The <code>double</code> value that is closer than any other to
     * <i>pi</i>, the ratio of the circumference of a circle to its
     * diameter.
     */
    public static final double PI = 3.14159265358979323846;

    /**
     * Returns the trigonometric sine of an angle.  Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the 
     * result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the sine of the argument.
     */
    public static double sin(double a) {
	//KML return StrictMath.sin(a); // default impl. delegates to StrictMath
    }
    
    /**
     * Returns the trigonometric cosine of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the 
     * result is NaN.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the cosine of the argument.
     */
    public static double cos(double a) {
	//KML return StrictMath.cos(a); // default impl. delegates to StrictMath
    }
   
    /**
     * Returns the trigonometric tangent of an angle.  Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the result 
     * is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   an angle, in radians.
     * @return  the tangent of the argument.
     */
    public static double tan(double a) {
	//KML return StrictMath.tan(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the arc sine of an angle, in the range of -<i>pi</i>/2 through
     * <i>pi</i>/2. Special cases: 
     * <ul><li>If the argument is NaN or its absolute value is greater 
     * than 1, then the result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   the value whose arc sine is to be returned.
     * @return  the arc sine of the argument.
     */
    public static double asin(double a) {
	//KML return StrictMath.asin(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the arc cosine of an angle, in the range of 0.0 through
     * <i>pi</i>.  Special case:
     * <ul><li>If the argument is NaN or its absolute value is greater 
     * than 1, then the result is NaN.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results 
     * must be semi-monotonic.
     *
     * @param   a   the value whose arc cosine is to be returned.
     * @return  the arc cosine of the argument.
     */
    public static double acos(double a) {
	//KML return StrictMath.acos(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the arc tangent of an angle, in the range of -<i>pi</i>/2
     * through <i>pi</i>/2.  Special cases: 
     * <ul><li>If the argument is NaN, then the result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   the value whose arc tangent is to be returned.
     * @return  the arc tangent of the argument.
     */
    public static double atan(double a) {
	//KML return StrictMath.atan(a); // default impl. delegates to StrictMath
    }

    /**
     * Converts an angle measured in degrees to an approximately
     * equivalent angle measured in radians.  The conversion from
     * degrees to radians is generally inexact.
     *
     * @param   angdeg   an angle, in degrees
     * @return  the measurement of the angle <code>angdeg</code>
     *          in radians.
     * @since   1.2
     */
    public static double toRadians(double angdeg) {
	return angdeg / 180.0 * PI;
    }

    /**
     * Converts an angle measured in radians to an approximately
     * equivalent angle measured in degrees.  The conversion from
     * radians to degrees is generally inexact; users should
     * <i>not</i> expect <code>cos(toRadians(90.0))</code> to exactly
     * equal <code>0.0</code>.
     *
     * @param   angrad   an angle, in radians
     * @return  the measurement of the angle <code>angrad</code>
     *          in degrees.
     * @since   1.2
     */
    public static double toDegrees(double angrad) {
	return angrad * 180.0 / PI;
    }

    /**
     * Returns Euler's number <i>e</i> raised to the power of a
     * <code>double</code> value.  Special cases:
     * <ul><li>If the argument is NaN, the result is NaN.
     * <li>If the argument is positive infinity, then the result is 
     * positive infinity.
     * <li>If the argument is negative infinity, then the result is 
     * positive zero.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   the exponent to raise <i>e</i> to.
     * @return  the value <i>e</i><sup><code>a</code></sup>, 
     *          where <i>e</i> is the base of the natural logarithms.
     */
    public static double exp(double a) {
	//KML return StrictMath.exp(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the natural logarithm (base <i>e</i>) of a <code>double</code>
     * value.  Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result 
     * is NaN.
     * <li>If the argument is positive infinity, then the result is 
     * positive infinity.
     * <li>If the argument is positive zero or negative zero, then the 
     * result is negative infinity.</ul>
     * <p>
     * A result must be within 1 ulp of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   a   a number greater than <code>0.0</code>.
     * @return  the value ln&nbsp;<code>a</code>, the natural logarithm of
     *          <code>a</code>.
     */
    public static double log(double a) {
	//KML return StrictMath.log(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the correctly rounded positive square root of a 
     * <code>double</code> value.
     * Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result 
     * is NaN. 
     * <li>If the argument is positive infinity, then the result is positive 
     * infinity. 
     * <li>If the argument is positive zero or negative zero, then the 
     * result is the same as the argument.</ul>
     * Otherwise, the result is the <code>double</code> value closest to 
     * the true mathematical square root of the argument value.
     * 
     * @param   a   a value.
     * <!--@return  the value of &radic;&nbsp;<code>a</code>.-->
     * @return  the positive square root of <code>a</code>.
     *          If the argument is NaN or less than zero, the result is NaN.
     */
    /*@ requires a >= 0.0;
      @ ensures \result * \result == a;
      @*/
    public static double sqrt(double a) {
	/*KML
	return StrictMath.sqrt(a); // default impl. delegates to StrictMath
				   // Note that hardware sqrt instructions
				   // frequently can be directly used by JITs
				   // and should be much faster than doing
				   // Math.sqrt in software.
				   */
    }

    /**
     * Computes the remainder operation on two arguments as prescribed 
     * by the IEEE 754 standard.
     * The remainder value is mathematically equal to 
     * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
     * where <i>n</i> is the mathematical integer closest to the exact 
     * mathematical value of the quotient <code>f1/f2</code>, and if two 
     * mathematical integers are equally close to <code>f1/f2</code>, 
     * then <i>n</i> is the integer that is even. If the remainder is 
     * zero, its sign is the same as the sign of the first argument. 
     * Special cases:
     * <ul><li>If either argument is NaN, or the first argument is infinite, 
     * or the second argument is positive zero or negative zero, then the 
     * result is NaN.
     * <li>If the first argument is finite and the second argument is 
     * infinite, then the result is the same as the first argument.</ul>
     *
     * @param   f1   the dividend.
     * @param   f2   the divisor.
     * @return  the remainder when <code>f1</code> is divided by
     *          <code>f2</code>.
     */
    public static double IEEEremainder(double f1, double f2) {
        //KML return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
    }

    /**
     * Returns the smallest (closest to negative infinity) 
     * <code>double</code> value that is not less than the argument and is 
     * equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a mathematical 
     * integer, then the result is the same as the argument. 
     * <li>If the argument is NaN or an infinity or positive zero or negative 
     * zero, then the result is the same as the argument. 
     * <li>If the argument value is less than zero but greater than -1.0, 
     * then the result is negative zero.</ul>
     * Note that the value of <code>Math.ceil(x)</code> is exactly the 
     * value of <code>-Math.floor(-x)</code>.
     *
     * @param   a   a value.
     * <!--@return  the value &lceil;&nbsp;<code>a</code>&nbsp;&rceil;.-->
     * @return  the smallest (closest to negative infinity) 
     *          floating-point value that is not less than the argument
     *          and is equal to a mathematical integer. 
     */
    public static double ceil(double a) {
	//KML return StrictMath.ceil(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the largest (closest to positive infinity) 
     * <code>double</code> value that is not greater than the argument and 
     * is equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a mathematical 
     * integer, then the result is the same as the argument. 
     * <li>If the argument is NaN or an infinity or positive zero or 
     * negative zero, then the result is the same as the argument.</ul>
     *
     * @param   a   a value.
     * <!--@return  the value &lfloor;&nbsp;<code>a</code>&nbsp;&rfloor;.-->
     * @return  the largest (closest to positive infinity) 
     *          floating-point value that is not greater than the argument
     *          and is equal to a mathematical integer. 
     */
    public static double floor(double a) {
	//KML return StrictMath.floor(a); // default impl. delegates to StrictMath
    }

    /**
     * Returns the <code>double</code> value that is closest in value
     * to the argument and is equal to a mathematical integer. If two
     * <code>double</code> values that are mathematical integers are
     * equally close, the result is the integer value that is
     * even. Special cases:
     * <ul><li>If the argument value is already equal to a mathematical 
     * integer, then the result is the same as the argument. 
     * <li>If the argument is NaN or an infinity or positive zero or negative 
     * zero, then the result is the same as the argument.</ul>
     *
     * @param   a   a <code>double</code> value.
     * @return  the closest floating-point value to <code>a</code> that is
     *          equal to a mathematical integer.
     */
    public static double rint(double a) {
	//KML return StrictMath.rint(a); // default impl. delegates to StrictMath
    }

    /**
     * Converts rectangular coordinates (<code>x</code>,&nbsp;<code>y</code>)
     * to polar (r,&nbsp;<i>theta</i>).
     * This method computes the phase <i>theta</i> by computing an arc tangent
     * of <code>y/x</code> in the range of -<i>pi</i> to <i>pi</i>. Special 
     * cases:
     * <ul><li>If either argument is NaN, then the result is NaN. 
     * <li>If the first argument is positive zero and the second argument 
     * is positive, or the first argument is positive and finite and the 
     * second argument is positive infinity, then the result is positive 
     * zero. 
     * <li>If the first argument is negative zero and the second argument 
     * is positive, or the first argument is negative and finite and the 
     * second argument is positive infinity, then the result is negative zero. 
     * <li>If the first argument is positive zero and the second argument 
     * is negative, or the first argument is positive and finite and the 
     * second argument is negative infinity, then the result is the 
     * <code>double</code> value closest to <i>pi</i>. 
     * <li>If the first argument is negative zero and the second argument 
     * is negative, or the first argument is negative and finite and the 
     * second argument is negative infinity, then the result is the 
     * <code>double</code> value closest to -<i>pi</i>. 
     * <li>If the first argument is positive and the second argument is 
     * positive zero or negative zero, or the first argument is positive 
     * infinity and the second argument is finite, then the result is the 
     * <code>double</code> value closest to <i>pi</i>/2. 
     * <li>If the first argument is negative and the second argument is 
     * positive zero or negative zero, or the first argument is negative 
     * infinity and the second argument is finite, then the result is the 
     * <code>double</code> value closest to -<i>pi</i>/2. 
     * <li>If both arguments are positive infinity, then the result is the 
     * <code>double</code> value closest to <i>pi</i>/4. 
     * <li>If the first argument is positive infinity and the second argument 
     * is negative infinity, then the result is the <code>double</code> 
     * value closest to 3*<i>pi</i>/4. 
     * <li>If the first argument is negative infinity and the second argument 
     * is positive infinity, then the result is the <code>double</code> value 
     * closest to -<i>pi</i>/4. 
     * <li>If both arguments are negative infinity, then the result is the 
     * <code>double</code> value closest to -3*<i>pi</i>/4.</ul>
     * <p>
     * A result must be within 2 ulps of the correctly rounded result.  Results
     * must be semi-monotonic.
     *
     * @param   y   the ordinate coordinate
     * @param   x   the abscissa coordinate
     * @return  the <i>theta</i> component of the point
     *          (<i>r</i>,&nbsp;<i>theta</i>)
     *          in polar coordinates that corresponds to the point
     *          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
     */
    public static double atan2(double y, double x) {
	//KML return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
    }

    /**
     * Returns the value of the first argument raised to the power of the
     * second argument. Special cases:
     *
     * <ul><li>If the second argument is positive or negative zero, then the 
     * result is 1.0. 
     * <li>If the second argument is 1.0, then the result is the same as the 
     * first argument.
     * <li>If the second argument is NaN, then the result is NaN. 
     * <li>If the first argument is NaN and the second argument is nonzero, 
     * then the result is NaN. 
     *
     * <li>If
     * <ul>
     * <li>the absolute value of the first argument is greater than 1
     * and the second argument is positive infinity, or
     * <li>the absolute value of the first argument is less than 1 and
     * the second argument is negative infinity,
     * </ul>
     * then the result is positive infinity. 
     *
     * <li>If 
     * <ul>
     * <li>the absolute value of the first argument is greater than 1 and 
     * the second argument is negative infinity, or 
     * <li>the absolute value of the 
     * first argument is less than 1 and the second argument is positive 
     * infinity,
     * </ul>
     * then the result is positive zero. 
     *
     * <li>If the absolute value of the first argument equals 1 and the 
     * second argument is infinite, then the result is NaN. 
     *
     * <li>If 
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is greater than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is less than zero,
     * </ul>
     * then the result is positive zero. 
     *
     * <li>If 
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is less than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is greater than zero,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If 
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is greater than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is less than zero but not a finite odd integer,
     * </ul>
     * then the result is positive zero. 
     *
     * <li>If 
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a positive finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a negative finite odd integer,
     * </ul>
     * then the result is negative zero. 
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is less than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is greater than zero but not a finite odd integer,
     * </ul>
     * then the result is positive infinity. 
     *
     * <li>If 
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a negative finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a positive finite odd integer,
     * </ul>
     * then the result is negative infinity. 
     *
     * <li>If the first argument is finite and less than zero
     * <ul>
     * <li> if the second argument is a finite even integer, the
     * result is equal to the result of raising the absolute value of
     * the first argument to the power of the second argument
     *
     * <li>if the second argument is a finite odd integer, the result
     * is equal to the negative of the result of raising the absolute
     * value of the first argument to the power of the second
     * argument
     *
     * <li>if the second argument is finite and not an integer, then
     * the result is NaN.
     * </ul>
     *
     * <li>If both arguments are integers, then the result is exactly equal 
     * to the mathematical result of raising the first argument to the power 
     * of the second argument if that result can in fact be represented 
     * exactly as a <code>double</code> value.</ul>
     * 
     * <p>(In the foregoing descriptions, a floating-point value is
     * considered to be an integer if and only if it is finite and a
     * fixed point of the method {@link #ceil <tt>ceil</tt>} or,
     * equivalently, a fixed point of the method {@link #floor
     * <tt>floor</tt>}. A value is a fixed point of a one-argument
     * method if and only if the result of applying the method to the
     * value is equal to the value.)
     *
     * <p>A result must be within 1 ulp of the correctly rounded
     * result.  Results must be semi-monotonic.
     *
     * @param   a   the base.
     * @param   b   the exponent.
     * @return  the value <code>a<sup>b</sup></code>.
     */
    public static double pow(double a, double b) {
	//KML return StrictMath.pow(a, b); // default impl. delegates to StrictMath
    }

    /**
     * Returns the closest <code>int</code> to the argument. The 
     * result is rounded to an integer by adding 1/2, taking the 
     * floor of the result, and casting the result to type <code>int</code>. 
     * In other words, the result is equal to the value of the expression:
     * <p><pre>(int)Math.floor(a + 0.5f)</pre>
     * <p>
     * Special cases:
     * <ul><li>If the argument is NaN, the result is 0.
     * <li>If the argument is negative infinity or any value less than or 
     * equal to the value of <code>Integer.MIN_VALUE</code>, the result is 
     * equal to the value of <code>Integer.MIN_VALUE</code>. 
     * <li>If the argument is positive infinity or any value greater than or 
     * equal to the value of <code>Integer.MAX_VALUE</code>, the result is 
     * equal to the value of <code>Integer.MAX_VALUE</code>.</ul> 
     *
     * @param   a   a floating-point value to be rounded to an integer.
     * @return  the value of the argument rounded to the nearest
     *          <code>int</code> value.
     * @see     java.lang.Integer#MAX_VALUE
     * @see     java.lang.Integer#MIN_VALUE
     */
    public static int round(float a) {
	return (int)floor(a + 0.5f);
    }

    /**
     * Returns the closest <code>long</code> to the argument. The result 
     * is rounded to an integer by adding 1/2, taking the floor of the 
     * result, and casting the result to type <code>long</code>. In other 
     * words, the result is equal to the value of the expression:
     * <p><pre>(long)Math.floor(a + 0.5d)</pre>
     * <p>
     * Special cases:
     * <ul><li>If the argument is NaN, the result is 0.
     * <li>If the argument is negative infinity or any value less than or 
     * equal to the value of <code>Long.MIN_VALUE</code>, the result is 
     * equal to the value of <code>Long.MIN_VALUE</code>. 
     * <li>If the argument is positive infinity or any value greater than or 
     * equal to the value of <code>Long.MAX_VALUE</code>, the result is 
     * equal to the value of <code>Long.MAX_VALUE</code>.</ul> 
     *
     * @param   a   a floating-point value to be rounded to a 
     *		<code>long</code>.
     * @return  the value of the argument rounded to the nearest
     *          <code>long</code> value.
     * @see     java.lang.Long#MAX_VALUE
     * @see     java.lang.Long#MIN_VALUE
     */
    public static long round(double a) {
	return (long)floor(a + 0.5d);
    }

    //KML private static Random randomNumberGenerator;

    private static synchronized void initRNG() {
        /*KML
	  if (randomNumberGenerator == null) 
            randomNumberGenerator = new Random();
	*/
    }

    /**
     * Returns a <code>double</code> value with a positive sign, greater 
     * than or equal to <code>0.0</code> and less than <code>1.0</code>. 
     * Returned values are chosen pseudorandomly with (approximately) 
     * uniform distribution from that range. 
     * <p>
     * When this method is first called, it creates a single new 
     * pseudorandom-number generator, exactly as if by the expression 
     * <blockquote><pre>new java.util.Random</pre></blockquote>
     * This new pseudorandom-number generator is used thereafter for all 
     * calls to this method and is used nowhere else. 
     * <p>
     * This method is properly synchronized to allow correct use by more 
     * than one thread. However, if many threads need to generate 
     * pseudorandom numbers at a great rate, it may reduce contention for 
     * each thread to have its own pseudorandom-number generator.
     *  
     * @return  a pseudorandom <code>double</code> greater than or equal 
     * to <code>0.0</code> and less than <code>1.0</code>.
     * @see     java.util.Random#nextDouble()
     */
    public static double random() {
        /*KML
	  if (randomNumberGenerator == null) initRNG();
	  return randomNumberGenerator.nextDouble();
	*/

    }

    /**
     * Returns the absolute value of an <code>int</code> value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned. 
     * <p>
     * Note that if the argument is equal to the value of 
     * <code>Integer.MIN_VALUE</code>, the most negative representable 
     * <code>int</code> value, the result is that same value, which is 
     * negative. 
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     * @see     java.lang.Integer#MIN_VALUE
     */
    public static int abs(int a) {
	return (a < 0) ? -a : a;
    }

    /**
     * Returns the absolute value of a <code>long</code> value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned. 
     * <p>
     * Note that if the argument is equal to the value of 
     * <code>Long.MIN_VALUE</code>, the most negative representable 
     * <code>long</code> value, the result is that same value, which is 
     * negative. 
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     * @see     java.lang.Long#MIN_VALUE
     */
    public static long abs(long a) {
	return (a < 0) ? -a : a;
    }

    /**
     * Returns the absolute value of a <code>float</code> value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the 
     * result is positive zero. 
     * <li>If the argument is infinite, the result is positive infinity. 
     * <li>If the argument is NaN, the result is NaN.</ul>
     * In other words, the result is the same as the value of the expression: 
     * <p><pre>Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(a))</pre>
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    public static float abs(float a) {
        return (a <= 0.0F) ? 0.0F - a : a;
    }
  
    /**
     * Returns the absolute value of a <code>double</code> value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the result 
     * is positive zero. 
     * <li>If the argument is infinite, the result is positive infinity. 
     * <li>If the argument is NaN, the result is NaN.</ul>
     * In other words, the result is the same as the value of the expression: 
     * <p><code>Double.longBitsToDouble((Double.doubleToLongBits(a)&lt;&lt;1)&gt;&gt;&gt;1)</code> 
     *
     * @param   a   the argument whose absolute value is to be determined
     * @return  the absolute value of the argument.
     */
    /*@ ensures \result == \real_abs(a);
      @*/
    public static double abs(double a) {
        return (a <= 0.0D) ? 0.0D - a : a;
    }

    /**
     * Returns the greater of two <code>int</code> values. That is, the 
     * result is the argument closer to the value of 
     * <code>Integer.MAX_VALUE</code>. If the arguments have the same value, 
     * the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of <code>a</code> and <code>b</code>.
     * @see     java.lang.Long#MAX_VALUE
     */
    /*@ ensures \result == \int_max(a,b);
      @*/
    public static int max(int a, int b) {
	return (a >= b) ? a : b;
    }

    /**
     * Returns the greater of two <code>long</code> values. That is, the 
     * result is the argument closer to the value of 
     * <code>Long.MAX_VALUE</code>. If the arguments have the same value, 
     * the result is that same value. 
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of <code>a</code> and <code>b</code>.
     * @see     java.lang.Long#MAX_VALUE
     */
    /*@ ensures \result == \int_max(a,b);
      @*/
    public static long max(long a, long b) {
	return (a >= b) ? a : b;
    }

    //KML private static long negativeZeroFloatBits = Float.floatToIntBits(-0.0f);
    //KML private static long negativeZeroDoubleBits = Double.doubleToLongBits(-0.0d);

    /**
     * Returns the greater of two <code>float</code> values.  That is,
     * the result is the argument closer to positive infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of <code>a</code> and <code>b</code>.
     */
    /*@ ensures \result == \real_max(a,b);
      @*/
    public static float max(float a, float b) {
        if (a != a) return a;	// a is NaN
	/*KML
	  if ((a == 0.0f) && (b == 0.0f)
	    && (Float.floatToIntBits(a) == negativeZeroFloatBits)) {
	    return b;
	}
	*/
	return (a >= b) ? a : b;
    }

    /**
     * Returns the greater of two <code>double</code> values.  That
     * is, the result is the argument closer to positive infinity. If
     * the arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the larger of <code>a</code> and <code>b</code>.
     */
    /*@ ensures \result == \real_max(a,b);
      @*/
    public static double max(double a, double b) {
        if (a != a) return a;	// a is NaN
	/*KML
	  if ((a == 0.0d) && (b == 0.0d)
	    && (Double.doubleToLongBits(a) == negativeZeroDoubleBits)) {
	    return b;
	}
	*/
	return (a >= b) ? a : b;
    }

    /**
     * Returns the smaller of two <code>int</code> values. That is,
     * the result the argument closer to the value of
     * <code>Integer.MIN_VALUE</code>.  If the arguments have the same
     * value, the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of <code>a</code> and <code>b</code>.
     * @see     java.lang.Long#MIN_VALUE
     */
    /*@ ensures \result == \int_min(a,b);
      @*/
    public static int min(int a, int b) {
	return (a <= b) ? a : b;
    }

    /**
     * Returns the smaller of two <code>long</code> values. That is,
     * the result is the argument closer to the value of
     * <code>Long.MIN_VALUE</code>. If the arguments have the same
     * value, the result is that same value.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of <code>a</code> and <code>b</code>.
     * @see     java.lang.Long#MIN_VALUE
     */
    /*@ ensures \result == \int_min(a,b);
      @*/
    public static long min(long a, long b) {
	return (a <= b) ? a : b;
    }

    /**
     * Returns the smaller of two <code>float</code> values.  That is,
     * the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero.  If
     * one argument is positive zero and the other is negative zero,
     * the result is negative zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of <code>a</code> and <code>b.</code>
     */
    /*@ ensures \result == \real_min(a,b);
      @*/
    public static float min(float a, float b) {
        if (a != a) return a;	// a is NaN
	/*KML
	  if ((a == 0.0f) && (b == 0.0f)
	    && (Float.floatToIntBits(b) == negativeZeroFloatBits)) {
	    return b;
	}
	*/
	return (a <= b) ? a : b;
    }

    /**
     * Returns the smaller of two <code>double</code> values.  That
     * is, the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other is negative zero, the
     * result is negative zero.
     *
     * @param   a   an argument.
     * @param   b   another argument.
     * @return  the smaller of <code>a</code> and <code>b</code>.
     */
    /*@ ensures \result == \real_min(a,b);
      @*/
    public static double min(double a, double b) {
        if (a != a) return a;	// a is NaN
	/*KML
	  if ((a == 0.0d) && (b == 0.0d)
	    && (Double.doubleToLongBits(b) == negativeZeroDoubleBits)) {
	    return b;
	}
	*/
	return (a <= b) ? a : b;
    }

}